Properties

Label 4608.2.a.y.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.874032\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{5} +3.16228 q^{7} +O(q^{10})\) \(q-1.41421 q^{5} +3.16228 q^{7} +4.47214 q^{11} +4.47214 q^{13} +6.32456 q^{17} -2.82843 q^{19} +4.00000 q^{23} -3.00000 q^{25} +4.24264 q^{29} -3.16228 q^{31} -4.47214 q^{35} +4.47214 q^{37} +6.32456 q^{41} -8.48528 q^{43} +12.0000 q^{47} +3.00000 q^{49} -7.07107 q^{53} -6.32456 q^{55} -13.4164 q^{61} -6.32456 q^{65} +8.00000 q^{71} -4.00000 q^{73} +14.1421 q^{77} +3.16228 q^{79} -4.47214 q^{83} -8.94427 q^{85} +14.1421 q^{91} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{23} - 12 q^{25} + 48 q^{47} + 12 q^{49} + 32 q^{71} - 16 q^{73} + 16 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 3.16228 1.19523 0.597614 0.801784i \(-0.296115\pi\)
0.597614 + 0.801784i \(0.296115\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.47214 1.34840 0.674200 0.738549i \(-0.264489\pi\)
0.674200 + 0.738549i \(0.264489\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.32456 1.53393 0.766965 0.641689i \(-0.221766\pi\)
0.766965 + 0.641689i \(0.221766\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264 0.787839 0.393919 0.919145i \(-0.371119\pi\)
0.393919 + 0.919145i \(0.371119\pi\)
\(30\) 0 0
\(31\) −3.16228 −0.567962 −0.283981 0.958830i \(-0.591655\pi\)
−0.283981 + 0.958830i \(0.591655\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.47214 −0.755929
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.32456 0.987730 0.493865 0.869539i \(-0.335584\pi\)
0.493865 + 0.869539i \(0.335584\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.07107 −0.971286 −0.485643 0.874157i \(-0.661414\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) −6.32456 −0.852803
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −13.4164 −1.71780 −0.858898 0.512148i \(-0.828850\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.32456 −0.784465
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.1421 1.61165
\(78\) 0 0
\(79\) 3.16228 0.355784 0.177892 0.984050i \(-0.443072\pi\)
0.177892 + 0.984050i \(0.443072\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.47214 −0.490881 −0.245440 0.969412i \(-0.578933\pi\)
−0.245440 + 0.969412i \(0.578933\pi\)
\(84\) 0 0
\(85\) −8.94427 −0.970143
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 14.1421 1.48250
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.5563 −1.54791 −0.773957 0.633238i \(-0.781726\pi\)
−0.773957 + 0.633238i \(0.781726\pi\)
\(102\) 0 0
\(103\) −9.48683 −0.934765 −0.467383 0.884055i \(-0.654803\pi\)
−0.467383 + 0.884055i \(0.654803\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.8885 −1.72935 −0.864675 0.502331i \(-0.832476\pi\)
−0.864675 + 0.502331i \(0.832476\pi\)
\(108\) 0 0
\(109\) 4.47214 0.428353 0.214176 0.976795i \(-0.431293\pi\)
0.214176 + 0.976795i \(0.431293\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −12.6491 −1.18993 −0.594964 0.803752i \(-0.702834\pi\)
−0.594964 + 0.803752i \(0.702834\pi\)
\(114\) 0 0
\(115\) −5.65685 −0.527504
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.0000 1.83340
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) 15.8114 1.40303 0.701517 0.712653i \(-0.252506\pi\)
0.701517 + 0.712653i \(0.252506\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.8885 1.56293 0.781465 0.623949i \(-0.214473\pi\)
0.781465 + 0.623949i \(0.214473\pi\)
\(132\) 0 0
\(133\) −8.94427 −0.775567
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.9737 −1.62103 −0.810515 0.585718i \(-0.800813\pi\)
−0.810515 + 0.585718i \(0.800813\pi\)
\(138\) 0 0
\(139\) −16.9706 −1.43942 −0.719712 0.694273i \(-0.755726\pi\)
−0.719712 + 0.694273i \(0.755726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.0000 1.67248
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.3848 −1.50614 −0.753070 0.657941i \(-0.771428\pi\)
−0.753070 + 0.657941i \(0.771428\pi\)
\(150\) 0 0
\(151\) 22.1359 1.80140 0.900699 0.434444i \(-0.143055\pi\)
0.900699 + 0.434444i \(0.143055\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.47214 0.359211
\(156\) 0 0
\(157\) 13.4164 1.07075 0.535373 0.844616i \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.6491 0.996890
\(162\) 0 0
\(163\) 19.7990 1.55078 0.775388 0.631485i \(-0.217554\pi\)
0.775388 + 0.631485i \(0.217554\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.07107 0.537603 0.268802 0.963196i \(-0.413372\pi\)
0.268802 + 0.963196i \(0.413372\pi\)
\(174\) 0 0
\(175\) −9.48683 −0.717137
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.8885 1.33705 0.668526 0.743689i \(-0.266925\pi\)
0.668526 + 0.743689i \(0.266925\pi\)
\(180\) 0 0
\(181\) 13.4164 0.997234 0.498617 0.866822i \(-0.333841\pi\)
0.498617 + 0.866822i \(0.333841\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.32456 −0.464991
\(186\) 0 0
\(187\) 28.2843 2.06835
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.07107 −0.503793 −0.251896 0.967754i \(-0.581054\pi\)
−0.251896 + 0.967754i \(0.581054\pi\)
\(198\) 0 0
\(199\) 3.16228 0.224168 0.112084 0.993699i \(-0.464247\pi\)
0.112084 + 0.993699i \(0.464247\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 13.4164 0.941647
\(204\) 0 0
\(205\) −8.94427 −0.624695
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.6491 −0.874957
\(210\) 0 0
\(211\) −5.65685 −0.389434 −0.194717 0.980859i \(-0.562379\pi\)
−0.194717 + 0.980859i \(0.562379\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 28.2843 1.90261
\(222\) 0 0
\(223\) −28.4605 −1.90586 −0.952928 0.303197i \(-0.901946\pi\)
−0.952928 + 0.303197i \(0.901946\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.4164 −0.890478 −0.445239 0.895412i \(-0.646881\pi\)
−0.445239 + 0.895412i \(0.646881\pi\)
\(228\) 0 0
\(229\) 13.4164 0.886581 0.443291 0.896378i \(-0.353811\pi\)
0.443291 + 0.896378i \(0.353811\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −16.9706 −1.10704
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.24264 −0.271052
\(246\) 0 0
\(247\) −12.6491 −0.804844
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −13.4164 −0.846836 −0.423418 0.905934i \(-0.639170\pi\)
−0.423418 + 0.905934i \(0.639170\pi\)
\(252\) 0 0
\(253\) 17.8885 1.12464
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.6491 0.789030 0.394515 0.918890i \(-0.370913\pi\)
0.394515 + 0.918890i \(0.370913\pi\)
\(258\) 0 0
\(259\) 14.1421 0.878750
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.3848 1.12094 0.560470 0.828175i \(-0.310621\pi\)
0.560470 + 0.828175i \(0.310621\pi\)
\(270\) 0 0
\(271\) −3.16228 −0.192095 −0.0960473 0.995377i \(-0.530620\pi\)
−0.0960473 + 0.995377i \(0.530620\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −13.4164 −0.809040
\(276\) 0 0
\(277\) 13.4164 0.806114 0.403057 0.915175i \(-0.367948\pi\)
0.403057 + 0.915175i \(0.367948\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.6491 0.754583 0.377291 0.926095i \(-0.376856\pi\)
0.377291 + 0.926095i \(0.376856\pi\)
\(282\) 0 0
\(283\) 28.2843 1.68133 0.840663 0.541559i \(-0.182166\pi\)
0.840663 + 0.541559i \(0.182166\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 20.0000 1.18056
\(288\) 0 0
\(289\) 23.0000 1.35294
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −29.6985 −1.73500 −0.867502 0.497434i \(-0.834276\pi\)
−0.867502 + 0.497434i \(0.834276\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 17.8885 1.03452
\(300\) 0 0
\(301\) −26.8328 −1.54662
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.9737 1.08643
\(306\) 0 0
\(307\) −11.3137 −0.645707 −0.322854 0.946449i \(-0.604642\pi\)
−0.322854 + 0.946449i \(0.604642\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.89949 −0.556011 −0.278006 0.960579i \(-0.589673\pi\)
−0.278006 + 0.960579i \(0.589673\pi\)
\(318\) 0 0
\(319\) 18.9737 1.06232
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −17.8885 −0.995345
\(324\) 0 0
\(325\) −13.4164 −0.744208
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 37.9473 2.09210
\(330\) 0 0
\(331\) −33.9411 −1.86557 −0.932786 0.360429i \(-0.882630\pi\)
−0.932786 + 0.360429i \(0.882630\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.1421 −0.765840
\(342\) 0 0
\(343\) −12.6491 −0.682988
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.47214 −0.240077 −0.120038 0.992769i \(-0.538302\pi\)
−0.120038 + 0.992769i \(0.538302\pi\)
\(348\) 0 0
\(349\) 4.47214 0.239388 0.119694 0.992811i \(-0.461809\pi\)
0.119694 + 0.992811i \(0.461809\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −25.2982 −1.34649 −0.673244 0.739420i \(-0.735100\pi\)
−0.673244 + 0.739420i \(0.735100\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.65685 0.296093
\(366\) 0 0
\(367\) 22.1359 1.15549 0.577743 0.816218i \(-0.303933\pi\)
0.577743 + 0.816218i \(0.303933\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −22.3607 −1.16091
\(372\) 0 0
\(373\) −22.3607 −1.15779 −0.578896 0.815401i \(-0.696516\pi\)
−0.578896 + 0.815401i \(0.696516\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.9737 0.977194
\(378\) 0 0
\(379\) 25.4558 1.30758 0.653789 0.756677i \(-0.273178\pi\)
0.653789 + 0.756677i \(0.273178\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) −20.0000 −1.01929
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.0416 1.21896 0.609480 0.792802i \(-0.291378\pi\)
0.609480 + 0.792802i \(0.291378\pi\)
\(390\) 0 0
\(391\) 25.2982 1.27939
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.47214 −0.225018
\(396\) 0 0
\(397\) −4.47214 −0.224450 −0.112225 0.993683i \(-0.535798\pi\)
−0.112225 + 0.993683i \(0.535798\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.9737 0.947500 0.473750 0.880659i \(-0.342900\pi\)
0.473750 + 0.880659i \(0.342900\pi\)
\(402\) 0 0
\(403\) −14.1421 −0.704470
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6.32456 0.310460
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.47214 0.218478 0.109239 0.994016i \(-0.465159\pi\)
0.109239 + 0.994016i \(0.465159\pi\)
\(420\) 0 0
\(421\) −40.2492 −1.96163 −0.980814 0.194948i \(-0.937546\pi\)
−0.980814 + 0.194948i \(0.937546\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18.9737 −0.920358
\(426\) 0 0
\(427\) −42.4264 −2.05316
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) −22.1359 −1.05649 −0.528245 0.849092i \(-0.677150\pi\)
−0.528245 + 0.849092i \(0.677150\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.3050 1.48734 0.743672 0.668545i \(-0.233083\pi\)
0.743672 + 0.668545i \(0.233083\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.32456 −0.298474 −0.149237 0.988801i \(-0.547682\pi\)
−0.149237 + 0.988801i \(0.547682\pi\)
\(450\) 0 0
\(451\) 28.2843 1.33185
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −20.0000 −0.937614
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.41421 0.0658665 0.0329332 0.999458i \(-0.489515\pi\)
0.0329332 + 0.999458i \(0.489515\pi\)
\(462\) 0 0
\(463\) 15.8114 0.734818 0.367409 0.930060i \(-0.380245\pi\)
0.367409 + 0.930060i \(0.380245\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.3607 1.03473 0.517364 0.855765i \(-0.326913\pi\)
0.517364 + 0.855765i \(0.326913\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −37.9473 −1.74482
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.82843 −0.128432
\(486\) 0 0
\(487\) −9.48683 −0.429889 −0.214945 0.976626i \(-0.568957\pi\)
−0.214945 + 0.976626i \(0.568957\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.8885 −0.807299 −0.403649 0.914914i \(-0.632258\pi\)
−0.403649 + 0.914914i \(0.632258\pi\)
\(492\) 0 0
\(493\) 26.8328 1.20849
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 25.2982 1.13478
\(498\) 0 0
\(499\) −16.9706 −0.759707 −0.379853 0.925047i \(-0.624026\pi\)
−0.379853 + 0.925047i \(0.624026\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 22.0000 0.978987
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.89949 0.438787 0.219394 0.975636i \(-0.429592\pi\)
0.219394 + 0.975636i \(0.429592\pi\)
\(510\) 0 0
\(511\) −12.6491 −0.559564
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.4164 0.591198
\(516\) 0 0
\(517\) 53.6656 2.36021
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.32456 0.277084 0.138542 0.990357i \(-0.455758\pi\)
0.138542 + 0.990357i \(0.455758\pi\)
\(522\) 0 0
\(523\) −14.1421 −0.618392 −0.309196 0.950998i \(-0.600060\pi\)
−0.309196 + 0.950998i \(0.600060\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.2843 1.22513
\(534\) 0 0
\(535\) 25.2982 1.09374
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 13.4164 0.577886
\(540\) 0 0
\(541\) −40.2492 −1.73045 −0.865225 0.501384i \(-0.832824\pi\)
−0.865225 + 0.501384i \(0.832824\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.32456 −0.270914
\(546\) 0 0
\(547\) −2.82843 −0.120935 −0.0604674 0.998170i \(-0.519259\pi\)
−0.0604674 + 0.998170i \(0.519259\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.0416 −1.01868 −0.509338 0.860566i \(-0.670110\pi\)
−0.509338 + 0.860566i \(0.670110\pi\)
\(558\) 0 0
\(559\) −37.9473 −1.60500
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.47214 0.188478 0.0942390 0.995550i \(-0.469958\pi\)
0.0942390 + 0.995550i \(0.469958\pi\)
\(564\) 0 0
\(565\) 17.8885 0.752577
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.9737 −0.795417 −0.397709 0.917512i \(-0.630195\pi\)
−0.397709 + 0.917512i \(0.630195\pi\)
\(570\) 0 0
\(571\) 22.6274 0.946928 0.473464 0.880813i \(-0.343003\pi\)
0.473464 + 0.880813i \(0.343003\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −12.0000 −0.499567 −0.249783 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.1421 −0.586715
\(582\) 0 0
\(583\) −31.6228 −1.30968
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −35.7771 −1.47668 −0.738339 0.674430i \(-0.764390\pi\)
−0.738339 + 0.674430i \(0.764390\pi\)
\(588\) 0 0
\(589\) 8.94427 0.368542
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −37.9473 −1.55831 −0.779155 0.626831i \(-0.784352\pi\)
−0.779155 + 0.626831i \(0.784352\pi\)
\(594\) 0 0
\(595\) −28.2843 −1.15954
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.7279 −0.517464
\(606\) 0 0
\(607\) −9.48683 −0.385059 −0.192529 0.981291i \(-0.561669\pi\)
−0.192529 + 0.981291i \(0.561669\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 53.6656 2.17108
\(612\) 0 0
\(613\) 13.4164 0.541884 0.270942 0.962596i \(-0.412665\pi\)
0.270942 + 0.962596i \(0.412665\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.2982 −1.01847 −0.509234 0.860628i \(-0.670071\pi\)
−0.509234 + 0.860628i \(0.670071\pi\)
\(618\) 0 0
\(619\) −11.3137 −0.454736 −0.227368 0.973809i \(-0.573012\pi\)
−0.227368 + 0.973809i \(0.573012\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 28.2843 1.12777
\(630\) 0 0
\(631\) 41.1096 1.63655 0.818274 0.574829i \(-0.194931\pi\)
0.818274 + 0.574829i \(0.194931\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.3607 −0.887357
\(636\) 0 0
\(637\) 13.4164 0.531577
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.32456 0.249805 0.124902 0.992169i \(-0.460138\pi\)
0.124902 + 0.992169i \(0.460138\pi\)
\(642\) 0 0
\(643\) −8.48528 −0.334627 −0.167313 0.985904i \(-0.553509\pi\)
−0.167313 + 0.985904i \(0.553509\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.41421 0.0553425 0.0276712 0.999617i \(-0.491191\pi\)
0.0276712 + 0.999617i \(0.491191\pi\)
\(654\) 0 0
\(655\) −25.2982 −0.988483
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) −4.47214 −0.173946 −0.0869730 0.996211i \(-0.527719\pi\)
−0.0869730 + 0.996211i \(0.527719\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.6491 0.490511
\(666\) 0 0
\(667\) 16.9706 0.657103
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −60.0000 −2.31627
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 38.1838 1.46752 0.733761 0.679408i \(-0.237763\pi\)
0.733761 + 0.679408i \(0.237763\pi\)
\(678\) 0 0
\(679\) 6.32456 0.242714
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.2492 1.54009 0.770047 0.637987i \(-0.220233\pi\)
0.770047 + 0.637987i \(0.220233\pi\)
\(684\) 0 0
\(685\) 26.8328 1.02523
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −31.6228 −1.20473
\(690\) 0 0
\(691\) 8.48528 0.322795 0.161398 0.986889i \(-0.448400\pi\)
0.161398 + 0.986889i \(0.448400\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26.8701 −1.01487 −0.507434 0.861691i \(-0.669406\pi\)
−0.507434 + 0.861691i \(0.669406\pi\)
\(702\) 0 0
\(703\) −12.6491 −0.477070
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −49.1935 −1.85011
\(708\) 0 0
\(709\) 13.4164 0.503864 0.251932 0.967745i \(-0.418934\pi\)
0.251932 + 0.967745i \(0.418934\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −12.6491 −0.473713
\(714\) 0 0
\(715\) −28.2843 −1.05777
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.7279 −0.472703
\(726\) 0 0
\(727\) −15.8114 −0.586412 −0.293206 0.956049i \(-0.594722\pi\)
−0.293206 + 0.956049i \(0.594722\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −53.6656 −1.98490
\(732\) 0 0
\(733\) −22.3607 −0.825911 −0.412955 0.910751i \(-0.635503\pi\)
−0.412955 + 0.910751i \(0.635503\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 45.2548 1.66473 0.832363 0.554231i \(-0.186988\pi\)
0.832363 + 0.554231i \(0.186988\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 0 0
\(745\) 26.0000 0.952566
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −56.5685 −2.06697
\(750\) 0 0
\(751\) 15.8114 0.576966 0.288483 0.957485i \(-0.406849\pi\)
0.288483 + 0.957485i \(0.406849\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −31.3050 −1.13930
\(756\) 0 0
\(757\) −40.2492 −1.46288 −0.731441 0.681904i \(-0.761152\pi\)
−0.731441 + 0.681904i \(0.761152\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.9737 0.687795 0.343897 0.939007i \(-0.388253\pi\)
0.343897 + 0.939007i \(0.388253\pi\)
\(762\) 0 0
\(763\) 14.1421 0.511980
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 29.6985 1.06818 0.534090 0.845428i \(-0.320654\pi\)
0.534090 + 0.845428i \(0.320654\pi\)
\(774\) 0 0
\(775\) 9.48683 0.340777
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.8885 −0.640924
\(780\) 0 0
\(781\) 35.7771 1.28020
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −18.9737 −0.677199
\(786\) 0 0
\(787\) 42.4264 1.51234 0.756169 0.654376i \(-0.227069\pi\)
0.756169 + 0.654376i \(0.227069\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −40.0000 −1.42224
\(792\) 0 0
\(793\) −60.0000 −2.13066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.3848 −0.651222 −0.325611 0.945504i \(-0.605570\pi\)
−0.325611 + 0.945504i \(0.605570\pi\)
\(798\) 0 0
\(799\) 75.8947 2.68496
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −17.8885 −0.631273
\(804\) 0 0
\(805\) −17.8885 −0.630488
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.9737 0.667079 0.333539 0.942736i \(-0.391757\pi\)
0.333539 + 0.942736i \(0.391757\pi\)
\(810\) 0 0
\(811\) 19.7990 0.695237 0.347618 0.937636i \(-0.386991\pi\)
0.347618 + 0.937636i \(0.386991\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −28.0000 −0.980797
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.7279 0.444208 0.222104 0.975023i \(-0.428708\pi\)
0.222104 + 0.975023i \(0.428708\pi\)
\(822\) 0 0
\(823\) 15.8114 0.551150 0.275575 0.961280i \(-0.411132\pi\)
0.275575 + 0.961280i \(0.411132\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −13.4164 −0.465971 −0.232986 0.972480i \(-0.574849\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.9737 0.657399
\(834\) 0 0
\(835\) −16.9706 −0.587291
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −11.0000 −0.379310
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.89949 −0.340553
\(846\) 0 0
\(847\) 28.4605 0.977914
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 17.8885 0.613211
\(852\) 0 0
\(853\) 13.4164 0.459369 0.229685 0.973265i \(-0.426231\pi\)
0.229685 + 0.973265i \(0.426231\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 44.2719 1.51230 0.756149 0.654399i \(-0.227078\pi\)
0.756149 + 0.654399i \(0.227078\pi\)
\(858\) 0 0
\(859\) 25.4558 0.868542 0.434271 0.900782i \(-0.357006\pi\)
0.434271 + 0.900782i \(0.357006\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.1421 0.479739
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 35.7771 1.20949
\(876\) 0 0
\(877\) −13.4164 −0.453040 −0.226520 0.974007i \(-0.572735\pi\)
−0.226520 + 0.974007i \(0.572735\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.6491 −0.426159 −0.213080 0.977035i \(-0.568349\pi\)
−0.213080 + 0.977035i \(0.568349\pi\)
\(882\) 0 0
\(883\) −19.7990 −0.666289 −0.333145 0.942876i \(-0.608110\pi\)
−0.333145 + 0.942876i \(0.608110\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) 50.0000 1.67695
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −33.9411 −1.13580
\(894\) 0 0
\(895\) −25.2982 −0.845626
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −13.4164 −0.447462
\(900\) 0 0
\(901\) −44.7214 −1.48988
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.9737 −0.630706
\(906\) 0 0
\(907\) 14.1421 0.469582 0.234791 0.972046i \(-0.424559\pi\)
0.234791 + 0.972046i \(0.424559\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −20.0000 −0.661903
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 56.5685 1.86806
\(918\) 0 0
\(919\) −34.7851 −1.14745 −0.573727 0.819047i \(-0.694503\pi\)
−0.573727 + 0.819047i \(0.694503\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 35.7771 1.17762
\(924\) 0 0
\(925\) −13.4164 −0.441129
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 56.9210 1.86752 0.933759 0.357903i \(-0.116508\pi\)
0.933759 + 0.357903i \(0.116508\pi\)
\(930\) 0 0
\(931\) −8.48528 −0.278094
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −40.0000 −1.30814
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 57.9828 1.89018 0.945092 0.326805i \(-0.105972\pi\)
0.945092 + 0.326805i \(0.105972\pi\)
\(942\) 0 0
\(943\) 25.2982 0.823823
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.8885 0.581300 0.290650 0.956830i \(-0.406129\pi\)
0.290650 + 0.956830i \(0.406129\pi\)
\(948\) 0 0
\(949\) −17.8885 −0.580687
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −18.9737 −0.614617 −0.307309 0.951610i \(-0.599428\pi\)
−0.307309 + 0.951610i \(0.599428\pi\)
\(954\) 0 0
\(955\) −28.2843 −0.915258
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −60.0000 −1.93750
\(960\) 0 0
\(961\) −21.0000 −0.677419
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 22.6274 0.728402
\(966\) 0 0
\(967\) −41.1096 −1.32200 −0.660998 0.750388i \(-0.729867\pi\)
−0.660998 + 0.750388i \(0.729867\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.3607 0.717588 0.358794 0.933417i \(-0.383188\pi\)
0.358794 + 0.933417i \(0.383188\pi\)
\(972\) 0 0
\(973\) −53.6656 −1.72044
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 56.9210 1.82106 0.910532 0.413439i \(-0.135672\pi\)
0.910532 + 0.413439i \(0.135672\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −33.9411 −1.07927
\(990\) 0 0
\(991\) −34.7851 −1.10498 −0.552492 0.833518i \(-0.686323\pi\)
−0.552492 + 0.833518i \(0.686323\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.47214 −0.141776
\(996\) 0 0
\(997\) 22.3607 0.708170 0.354085 0.935213i \(-0.384792\pi\)
0.354085 + 0.935213i \(0.384792\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4608.2.a.y.1.2 yes 4
3.2 odd 2 4608.2.a.x.1.4 yes 4
4.3 odd 2 4608.2.a.x.1.1 4
8.3 odd 2 4608.2.a.x.1.3 yes 4
8.5 even 2 inner 4608.2.a.y.1.4 yes 4
12.11 even 2 inner 4608.2.a.y.1.3 yes 4
16.3 odd 4 4608.2.d.l.2305.4 4
16.5 even 4 4608.2.d.i.2305.1 4
16.11 odd 4 4608.2.d.l.2305.2 4
16.13 even 4 4608.2.d.i.2305.3 4
24.5 odd 2 4608.2.a.x.1.2 yes 4
24.11 even 2 inner 4608.2.a.y.1.1 yes 4
48.5 odd 4 4608.2.d.l.2305.3 4
48.11 even 4 4608.2.d.i.2305.4 4
48.29 odd 4 4608.2.d.l.2305.1 4
48.35 even 4 4608.2.d.i.2305.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.x.1.1 4 4.3 odd 2
4608.2.a.x.1.2 yes 4 24.5 odd 2
4608.2.a.x.1.3 yes 4 8.3 odd 2
4608.2.a.x.1.4 yes 4 3.2 odd 2
4608.2.a.y.1.1 yes 4 24.11 even 2 inner
4608.2.a.y.1.2 yes 4 1.1 even 1 trivial
4608.2.a.y.1.3 yes 4 12.11 even 2 inner
4608.2.a.y.1.4 yes 4 8.5 even 2 inner
4608.2.d.i.2305.1 4 16.5 even 4
4608.2.d.i.2305.2 4 48.35 even 4
4608.2.d.i.2305.3 4 16.13 even 4
4608.2.d.i.2305.4 4 48.11 even 4
4608.2.d.l.2305.1 4 48.29 odd 4
4608.2.d.l.2305.2 4 16.11 odd 4
4608.2.d.l.2305.3 4 48.5 odd 4
4608.2.d.l.2305.4 4 16.3 odd 4